Midwest Topology Seminar, Winter/Spring 2021

The Midwest Topology Seminar in Winter/Spring 2021 will be held in an online format.  Three 45-minute talks will be separated by two 30-minute coffee breaks.  The breaks will be structured to provide an opportunity for small group conversation in breakout rooms.  Participants will be able to move themselves between rooms, and will be able to see who is in other rooms. (Note: Breakout room functionality works imperfectly with a smartphone or tablet.  It’s best to use a computer for this.)

Schedule

Thursday April 8 (all times eastern)

Registration

Registration is not required to participate.

Abstracts

Speaker: Jeremy Hahn
Title: Connections between the Segal and Lichtenbaum-Quillen conjectures

Abstract: I will describe how ideas of Sverre Lunøe-Nielsen, Akhil Mathew, John Rognes, Dylan Wilson, and others connect the Segal and Lichtenbaum-Quillen conjectures. I will also indicate how the Segal conjecture has been proved for the topological Hochschild homologies of many ring spectra, often implying Lichtenbaum-Quillen style results.

Speaker: Kate Ponto
Title: Homotopical invariants for lifting and extensions

Abstract: Motivated by questions in intersection theory, Klein and Williams defined a homotopical obstruction to lifts. While they considered topological examples, this approach is remarkably flexible. I’ll describe how to make sense of this invariant in categories with homotopies and a Blakers-Massey theorem.

Speaker: Kristine Bauer
Title: Categorical differentiation of homotopy functors

Abstract: The Goodwillie functor calculus tower is an approximation of a homotopy functor which resembles the Taylor series approximation of a function in ordinary calculus. In 2017, B., Johnson, Osborne, Tebbe and Riehl (BJORT, collectively) showed that the directional derivative for functors of an abelian category are an example of a categorical derivative in the sense of Blute, Cockett and Seely. The BJORT result relied on the fact that the target and source of the functors in question were both abelian categories. This leads one to the question of whether or not other sorts of homotopy functors have a similar structure.

To address this question, Burke and Ching and I instead use the notion of tangent categories, due to Rosicky, Cockett-Cruttwell and via an incarnation due to Leung. The structure of a tangent category is highly reminiscent of the structure of a tangent bundle on a manifold. Indeed, the category of smooth manifolds is a primary and motivating example of a tangent category. In recent work with Burke and Ching, we make precise the notion of a tangent infinity category, and show that the directional derivative for homotopy functors appears as the associated categorical derivative of a particular tangent infinity category. This ties together Lurie’s tangent bundle construction to the categorical literature on tangent categories.

In this talk, I aim to explain the categorical notions of differentiation and tangent categories, and explain their relationship to Goodwillie’s functor calculus.

Organizers

Mark Behrens, Robert Bruner, Paul Goerss, Dan Isaksen, and Vesna Stojanoska

Previous seminars

Spring 2020 Midwest Topology Seminar

Fall 2020 Midwest Topology Seminar